What kind of work do modern day algebraists do?

I don't believe what I'm doing is especially active or popular (so hopefully someone else will respond with a better answer), but seeing as no one has answered yet, I'll just mention one of the things algebraists do: invent new algebras.

The process is very easy to describe. It may or may not result in something useful. Take a set $A$ and define a set $F$ of operations on $A$ (maps from $A^n$ into $A$, for various non-negative integer values of $n$). The set $A$ plus the operations $F$ is what we call an algebra, usually denoted $\mathbf{A} = \langle A, F\rangle$. The algebras you already know (e.g., groups, rings, modules) are examples.

In my work, I think about different ways to construct such algebras. Usually I work with finite algebras, often using computer software like GAP or the Universal Algebra Calculator to construct examples and study them. I look at the important features of the algebras and try to understand them better and make general statements about them.

To address your last question, there is the following open problem that I worked on as a graduate student: Given a finite lattice $L$, does there exist a finite algebra $\mathbf{A}$ (as described above) such that $L$ is the congruence lattice of $\mathbf{A}$. This question is at least 50 years old and quite important for our understanding of finite algebras. In 1980 it was discovered (by Palfy and Pudlak) to be equivalent to the following open problem about finite groups: given a finite lattice $L$, can we always find a finite group that has $L$ as an interval in its subgroup lattice? Imho, these are fun problems to work on.

A pure ring theorist will often have thought quite a bit about Kothe's conjecture. They will have thought about stuff that has the morpheme "nil" in it. Is the polynomial ring of a nil ring nil? Nilpotent maybe? When you hear these theorems and problems for the first time in a single talk, you might have trouble distinguishing between open questions and solved or even trivial problems soon after that. They all sound rather similar. In general, there are loads of simple-sounding problems like Kothe's conjecture in ring theory that are difficult. Many of them have probably never been asked.

Some of ring theorists, I believe these are mainly from Iran, will have considered some kind of graph defined by ring-theoretic stuff. Take a ring and call the zero divisors vertices. Throw in an edge between $x$ and $y$ whenever $xy=0$. You get a graph that you can do all kinds of things with. You can ask which rings induce a graph with this or that property.

Often a noncommutative ring theorist will be looking for some kinds of left-right symmetries. If you define a left Xical ring-theoretic thing and a right Xical ring-theoretic thing, are they the same ring-theoretic thing? The Jacobson radical may have been the inspiration for this.

Another thing is that, as in a lot of mathematics, algebraists will be trying to classify their objets. Rings in general don't seem reasonably classifiable, which leaves room for attempts at partial classification. In ring/algebra theory these will often aim at generalizing Wedderburn's theorem.

This is a very narrow part of what algebraists do. Algebra comes in so many flavors. The commutative-noncommutative boundary is especially strong I think. Also, some algebraists will think a lot about universal algebra, varieties and pseudo-varieties, some won't. Some will be deeply in love in categories, some will say meh.